Aerospace Science and Technology 32 (2014) 35–41

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Aerospace Science and Technology

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Autonomous orbit determination and its error analysis for deep spaceusing X-ray pulsar

Dongzhu Feng a, Hehe Guo b,∗, Xin Wang c, Xiaoguang Yuan a

a School of Electronic Engineering, Xidian University, Xi’an, 710071, Chinab Space Star Technology Co., Ltd., Beijing, 100086, Chinac College of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 January 2013Received in revised form 12 December 2013Accepted 12 December 2013Available online 2 January 2014

Keywords:Orbit determinationOrbit mutationOrbit initial errorX-ray pulsarAUKF

Autonomous orbit determination (OD) is a complex process using filtering method to integrateobservation and orbit dynamic model effectively and estimate the position and velocity of a spacecraft.As a novel technology for autonomous interplanetary OD, X-ray pulsar holds great promise for deepspace exploration. The position and velocity of spacecraft should be estimated accurately during the ODprocess. However, under the same condition, the accuracy of OD can be greatly reduced by the error ofthe initial orbit value and the orbit mutation. To resolve this problem, we propose a novel OD method,which is based on the X-ray pulsar measurement and Adaptive Unscented Kalman Filter (AUKF). Theaccuracy of OD can be improved obviously because the AUKF estimates the orbit of spacecraft usingmeasurement residual. During the simulation, the orbit of Phoenix Mars Lander, Deep Impact Probe,and Voyager 1 are selected. Compared with Unscented Kalman Filter (UKF) and Extended Kalman Filter(EKF), the simulation results demonstrate that the proposed OD method based on AUKF can accuratelydeterminate the velocity and position and effectively decrease the orbit estimated errors which is causedby the orbit mutation and orbit initial errors.

Crown Copyright © 2014 Published by Elsevier Masson SAS. All rights reserved.

1. Introduction

Orbit determination (OD) of spacecrafts is a critical and essen-tial technology which determines the success or failure of the deepspace exploration. It is well known that the OD systems of deepspace exploration is predominated by measurements of range andrange rate information obtained as part of the uplink and downlinkcommunication with ground stations such as NASA’s Deep SpaceNetwork (DSN) [16]. The DSN system is required at least three sta-tions, and every station requires huge antennas, ultra-sensitive re-ceivers, and powerful transmitters in order to transmit and receiveover the vast distances involved. Therefore it is expensive to buildand maintain the DSN to support the exploration, and its highcost, lack of robustness to loss of contact, space segment degrada-tion, and other factors promote the application of autonomous ODsystem, which is less costly and less vulnerable in hostile environ-ment [13,14]. With the development of deep space exploration, therapid increasing spacecrafts has resulted in a continuing cost onoperating spacecrafts. Consequently, autonomous OD system andnavigation system are highly attractive due to a reduction in oper-ating costs and less vulnerable. It was reported by NASA that X-ray

* Corresponding author. Tel.: +86 135 2167 1569.E-mail addresses: dongzhufeng@gmail.com (D. Feng), xiaohe5202@163.com

(H. Guo), wangxinnwpu@gmail.com (X. Wang), xgyuan@xidian.edu.cn (X. Yuan).

1270-9638/$ – see front matter Crown Copyright © 2014 Published by Elsevier Massonhttp://dx.doi.org/10.1016/j.ast.2013.12.008

pulsars would be a kind of new methods in the future next 25years for deep space autonomous OD system [11].

Pulsars are rapidly rotating neutron stars, which producehighly regular and very stable pulses that can be predicted accu-rately [16]. Pulsars are like the beacons in the universe, thereforethe significant research efforts have been undertaken in pulsarnavigation area. In recent year, pulsar navigation attracts more at-tention in many research groups such as the Defense AdvancedResearch Projects Agency of the United States, the European SpaceAgency and China [10]. Pulsed signals, emitted by X-ray continu-ally, can be detected by X-ray sensors which are placed onboardspacecraft [20]. These pulsed signals are processed to obtain apulse Time of Arrival (TOA) at the spacecraft [21]. Using the mea-sured pulse TOAs from variable X-ray pulsars and an optimal stateestimation technology, the spacecraft’s orbit can be determinedaccurately. During the OD, the corresponding time at the Solar Sys-tem Barycenter (SSB) can be provided by the pulsar timing model[19]. The use of X-ray pulsars for spacecraft navigation and OD indeep space exploration has been considered ever since their dis-covery [9]. P.S. Ray [16], T.J. Martin-Mur [11], and P. Graven [6] ad-dressed the use of X-ray Pulsar Navigation (XPNAV) in deep spaceexploration in 2008. P.S. Ray et al. [16] addressed the possibil-ity of the use of XPNAV in deep space exploration. J. Liu et al. [10]established the Constant Velocity (CV) model, and utilized the aug-mented state unscented Kalman filter to estimate the spacecraft’s

SAS. All rights reserved.

36 D. Feng et al. / Aerospace Science and Technology 32 (2014) 35–41

Fig. 1. Orbit of deep impact.

Fig. 2. Orbit of Voyager 1 encounter with Jupiter and Saturn.

positions. The accuracy of navigation was improved in the presenceof the clock error and the pulsar direction error for the interplan-etary missions. Furthermore, Ref. [8] also analyzed the impact ofthe pulsar direction error on the navigation system, and proposedan ASUKF-based navigation method that mitigated the impact onthe performance of the navigation system and improved the nav-igation accuracy significantly. E. Wei et al. [23] investigated anautonomous navigation for Mars probe orbit based on X-ray pul-sars. Compared with the VLBI or Doppler positioning solution, theX-ray pulsar-based position estimation of Mars probe had higheraccuracy. The results of Ref. [23] identified the feasibility of usingthe X-ray pulsars for autonomous navigation of the Mars probe or-bit. However the orbit was limited to the near-Mars orbit. It canbe seen from above discussions that the orbit characteristics ofspacecraft, as orbit mutation and orbit initial error, have not beenaddressed in detail in previous studies. Therefore, our paper is de-voted to analyze the OD system based on the X-ray pulsars withrespect to the orbital characteristics.

In deep space exploration, the spacecraft orbit can changegreatly in a short time due to the uncertainties of orbit dynamicmodel and the abnormal disturbance of unknown planetary, suchas comets and the satellites of planetary [12]. This phenomenonis illustrated as the cruise phase orbit of Deep Impact and the or-bit Voyager 1 probes encounter with Jupiter and Saturn in Fig. 1and Fig. 2. The cruise phase of Deep Impact is extremely smooth,while in the phase of encountering with Jupiter and Saturn, theorbit of Voyager 1 changes greatly due to the gravity of Jupiter,Saturn, and their small satellites. As the pulsar signal is extremelyweak, it takes a long time to accumulate the pulse profile to ob-tain a higher accuracy of the measurement value. Thus, it usuallytakes 10 minutes for the pulsar navigation system to filter. Whenthe orbit changes greatly in a short time or larger errors exist inthe orbit initial value, errors will appear in the OD of spacecraft.The accumulation of errors will lead to the deviation of the esti-mated value from the true value, and the divergence of filteringalgorithm. In order to decrease the error of OD, we propose an

Fig. 3. Ranging principle of X-ray pulsar.

Adaptive Unscented Kalman Filter (AUKF) algorithm to determi-nate the orbit based on the X-ray pulsar. The observation equationis developed relying on the X-ray pulsar’s TOA. The state equa-tion of each spacecraft is established according to its orbit model.Then the position and velocity of spacecraft are estimated by AUKF,which uses measurement residual value to decrease the impact ofinitial errors and orbit mutation on the OD accuracy. The simula-tion results are evaluated using several typical filters and recordedflight data from NASA website.

This paper is organized as follows. After this introduction, theprinciple of OD system using X-ray pulsar, the measurement modelbased on X-ray and orbit dynamic model of deep space probe areintroduced in Section 2. Then orbit estimator based on AUKF isdescribed in Section 3. In Section 4, the simulation conditions andthe simulation results are shown for illustrating the effectivenessof the proposed AUKF for the OD system based on X-ray pulsar. InSection 5, conclusions close the paper.

2. Measurement model and orbit dynamic model

2.1. Measurement model based on X-ray pulsar

X-ray pulsars can provide a kind of unique pulse signals whichcan be detected by X-ray sensors. These signals are processed toobtain an observed pulsar profile with a high signal-to-noise ra-tio after a period of observation about 5–10 minutes. The principleof using X-ray pulsar for spacecraft autonomous OD is compar-ing the observed pulsar profile by X-ray sensor at the spacecraftwith the standard pulsar profile at the SSB to determine residualphase within a cycle. By computing the phase and the integer cy-cle ambiguity, we can get the time delay of the pulse from the SSBto spacecraft along the direction of pulsar angular position vector[18,22].

The diagram of Fig. 3 shows that arriving pulses are used toupdate the spacecraft position with respect to the BarycentricCelestial Reference System (BCRS). O SSB is the origin of coordi-nates. XSSB points to mean equator at the standard epoch J2000.0.O SSB-XSSBYSSB ZSSB is SSB coordinate system constructed in Celes-tial equatorial plane. PSRi denotes the ith pulsar. SC is spacecraft.hi represents angular position vector of the ith pulsar. r meansthe position vector of spacecraft in SSB coordinate system. Fromthe geometry of O SSB , PSRi and SC, the projection of spacecraft po-sition vector in pulsar united angular position vector is

zi = hTi r = c(tSSB − tsc) (1)

where tSSB is TOA of the N th pulsar photon at the O SSB , tsc

is TOA of spacecraft, c indicates the speed of light, and hi =[cosλ cosα, cosλ sinα, sinλ] respectively, here α represents theright ascension of pulsar in the celestial coordinate, and λ repre-sents the declination of pulsar in the celestial coordinate. The timedelay �t between tSSB and tsc is given as [4]

D. Feng et al. / Aerospace Science and Technology 32 (2014) 35–41 37

�t = tSSB − tsc

= hi · r

c+ 1

2cD0

[(hi · r)2 − r2 + 2(hi · b)(hi · b) − 2(b · r)

]+ 2μs

c3ln

∣∣∣∣ hi · r + r

hi · b + b+ 1

∣∣∣∣ (2)

where b is the position vector of O SSB relative to the sun, D0 isthe range from the pulsar to the SSB.

If zi is selected as measurement vector, then zi can be ex-pressed as zi = c�ti . Assuming m pulsars are used for OD, themeasurement equation [5] can be presented as

z = h(

X(k)) + v(k) (3)

where z = [z1, z2, . . . , zm]T , and v(k) is measurement noise, whichcan be calculated by signal-to-noise ratio of observed pulsar signalwith covariance R .

2.2. Orbit dynamic model

As spacecraft force is complex in deep space, we use the sim-ple two-body motion, the principal orbit motion, in the orbitaldynamics model. In space, the most bodies’ orbits can be de-scribed as two-body orbits to a fair degree of accuracy. Orbitalelements are the parameters and states which completely spec-ify the basic two-body motion of orbit bodies. In this set, the orbitis determined by position and velocity [15]. During the interplan-etary travel, the spacecraft orbit changes greatly in a short timedue to another planetary gravity when they encounter. In order toestimate state variable of spacecraft as accurately as possible, sunradiation pressure and third body gravitational perturbations aretaken into account. State vector x is composed of position vectorr = [x, y, z]T and velocity vector v = [vx, v y, vz]T in the heliocen-tric inertial coordinate system, the common state equation of ODsystem is give as{

drdt = v + w1

dvdx = asun + a3-body + asr + w2

(4)

where w1 means the system position noise, w2 means the systemvelocity noise.

asun = −μsrr3 is the gravitational acceleration of the Sun, μs is

the gravitational constant of the Sun.a3-body = ∑n

i=1 μi(ri−r

‖ri−r‖3 − ri‖ri‖3 ) is the gravitational acceler-

ation of other planet, μi represents the gravitational constant ofthe ith planet, ri is the position vector of the ith planet in theheliocentric inertial coordinates, which is obtained from the JPLephemeris.

asr = −kP sSm (1AU )2 r

r3 is the Sun pressure acceleration, k is thereflection coefficient, and is influenced by the surface material,roughness and shape of spacecraft, P s denotes the compressivestrength of sunlight, S/m is the effective area mass ration in termsof solar radiation pressure, and r = √

x2 + y2 + z2 is the distancebetween the spacecraft and sun.

Eq. (4) can be written in the general state model as

x(k) = F(x(k)

) + w(k) (5)

where w(k) = [w1, w2]T is the state process noise, and can beassumed as the zero-mean white noise with covariance Q .

3. Orbit estimators

Eqs. (3) and (5) are the measurement model and system statemodel respectively associated with particular orbiting spacecraftand data tracking system. Here x is the vector of n-states of the

chosen coordinate system and other augmented parameters, z isthe vector of observations, F , h are known, nonlinear functionalrelationships, and w , v are the process and measurement noisevectors.

We next elucidate important features of some best known esti-mation algorithms to obtain the filtered or smoothed estimates ofthe orbital states of the spacecraft. These algorithms will be corpo-rate into filtering framework based on the model form of Eqs. (3)and (5) as follows:

xk = F (xk−1,k − 1) + wk

zk = h(xk,k) + vk (6)

where xk−1 is the vector of n-states of the chosen coordinate sys-tem, denotes the system state at time k − 1, zk denotes the ob-servations vector at time k, wk denotes the process noise, and vkdenotes the measurement noise. The F and h represent the stateand measurement models.

The Extended Kalman Filter (EKF) is the minimum mean-square-error estimator based on the Taylor series expansion ofthe nonlinear functions [14], while the Unscented Kalman Filter(UKF) utilizes the unscented transformation to capture the meanand covariance estimates with a minimal set of sample points [7].Therefore, UKF is more suitable to nonlinear filtering in practicalapplications in that it can overcome the shortcomings of the EKF.But UKF is more sensitive to the initial value, the errors of whichwill directly affect the OD accuracy. Even if the initial value is ac-curate, some deviations exist in the estimates obtained from UKFbecause of the orbit mutation, which is caused by uncertainties ofthe orbit dynamic model and the presence of abnormal perturba-tion error. This paper apply AUKF to deal with the inadequacies ofOD system based on X-ray pulsar for deep space, such as abnormaldisturbance, orbit mutation, long filtering cycle, initial error and soon. On the basis of UKF algorithm, the AUKF algorithm uses real-time observational information to adjust UKF estimate, and makeserror covariance matrix of state variable and measurement adap-tive expansion by utilizing measurement residual information, andadopting the principle of variance inflation.

The steps of AUKF are as follows.(1) Initialize

x0 = E(x0), P 0 = E[(x0 − x0)(x0 − x0)

T ](7)

where P is initial error covariance of state vector.(2) When k = 0,1,2,3, . . . , calculate sigma points

χk = [xk xk + √

n + λ√

(P k)i xk − √n + λ

√(P k)i

](8)

where i = 1,2, . . . ,n , λ = α2(n +κ)−n, α is the distance betweensigma points and xk , general 10−4 � α � 1, κ is 0 or 3 − n, n isnumber of state vector; when P k = A AT,

√(P k)i represents the

ith line of A.(3) Time update

χ i,k+1 = F (χ i,k,k) (9)

x−k+1 =

2n∑i=0

ωmi χ i,k+1 (10)

P −k+1 =

2n∑i=0

ωci

[χ i,k+1 − x−

k+1

][χ i,k+1 − x−

k+1

]T + Q (11)

where x−k+1 is the prediction value of orbit, P −

k+1 is error covari-ance matrix, Q means process noise covariance matrix; wm

i , wci

are the weighted coefficients of mean and covariance, and wm0 =

λ/(n + λ), wc0 = λ/(n + λ) + (1 −α2 + β), wm

i = wci = 1/[2(n + λ)],

i = 1,2, . . . ,2n , for Gaussian x, the optimal value is β = 2.

38 D. Feng et al. / Aerospace Science and Technology 32 (2014) 35–41

Table 1The start time of selected orbit phase.

Probe Orbit Start time End time

Phoenix Cruise phase 2007.10.15.00:00:00.0000 2007.10.22.00:00:00.0000Deep Impact Cruise phase 2005.03.15.00:00:00.0000 2005.03.22.00:00:00.0000Voyager 1 Encounter with Jupiter phase 1979.03.02.00:00:00.0000 1979.03.09.00:00:00.0000

Encounter with Saturn phase 1980.11.11.00:00:00.0000 1980.11.20.00:00:00.0000

(4) Measurement update

zi,k+1 = h(χ i,k+1,k + 1) (12)

z−k+1 =

2n∑i=0

ωmi zi,k+1 (13)

P zk+1 zk+1= 1

ak

2n∑i=0

ωci

[zi,k+1 − z−

k+1

][zi,k+1 − z−

k+1

]T + Rk+1

(14)

P xk+1 zk+1= 1

ak

2n∑i=0

ωci

[χ i,k+1 − x−

k+1

][zi,k+1 − z−

k+1

]T(15)

K k+1 = P xk+1 zk+1P −1

zk+1 zk+1(16)

P k+1 = 1

akP −

k+1 − K k+1 P zk+1 zk+1K T

K+1 (17)

xk+1 = x−k+1 + K k+1

(zk+1 − z−

k+1

)(18)

where Rk+1 is measurement noise covariance matrix, ak is adap-tive factor [24], and 0 < ak � 1. In UKF algorithm, the observed in-formation z−

k+1 presents a large deviation relative to the true valuezi,k+1, because of the uncertainty of the orbit dynamics model, theerror of initial value and long filter cycle. To further reduce theimpact of these factors on orbit accuracy, adaptive expansion isapplied in the covariance P zk+1 zk+1

, P xk+1 zk+1, P k+1 thus reducing

the influence of the error of observation information z−k+1 on the

gain K k+1. So, ak is constructed by estimated covariance matrix ofmeasurement residuals information V k+1. We get adaptive factoras

ak ={

1 tr(V k+1k V Tk+1) � tr(P )

tr(P )

tr(V k+1 V Tk+1)

tr(V k+1 V Tk+1) > tr(P )

(19)

where measurement residuals V k+1 = zk+1 − ∑2ni=0 ωm

i zi,k+1, P =∑2ni=0 ωc

i [zi,k+1 − z−k+1][zi,k+1 − z−

k+1]T .From above steps, we can draw that the adaptive factor ak is

equal to different value at different conditions. The adaptive factorak is less than 1 when the value of orbital dynamics model pre-diction exist deviations. It means that the impact of initial valueand orbit characteristic on orbit prediction is smaller. The adap-tive factor ak will be close to 0 when the deviations are bigger. Itmeans that the predicted orbit value, which is obtained from orbitdynamic model, is completely discarded. By Eqs. (14)–(17), it is ob-vious that the factor ak can adaptively reduce the impact of orbitdynamic model error on OD accuracy, and thus reduce the impactcaused by initial error and bigger orbit change on OD accuracy byusing observed information based on measurement residuals V k .

4. Simulations

4.1. Simulation conditions

In order to demonstrate the effectiveness of the proposed ODmethod, we select the orbits of Phoenix Mars Lander, Deep ImpactProbe, and Voyager 1 as three typical spacecrafts, and compare

Table 2Parameters and range measurements of pulsars applied for OD.

Pulsar α (J2000) δ (J2000) D0 Range measurement accuracy (m)

B0531+21 83.63 22.01 2.0 109B1821−24 276.13 −24.87 5.5 325B1937+21 294.92 21.58 3.6 344

them with the EKF, UKF, AUKF orbit estimators. The selected orbitsegments and start and end times of Phoenix [17], Deep Impact [2]and Voyager 1 [3] are shown in Table 1. The initial value and trueorbit position vector of these spacecrafts are obtained from JPL’sHorizon System [1] with an interval of 600 s. As the Voyager 1spacecraft is far away from the Sun and its forces are more com-plex, the gravitational perturbations from moons of Jupiter, moonsof Saturn and other asteroids are not taken into account duringVoyager 1 encounter with Jupiter and Saturn.

The simulation parameters are as follows:

(a) The sampling period is 600 s, and the used pulsars andtheir parameters are shown in Table 2. [Observing condi-tions: Area = 1 m2, X-ray background = 0.005 ph/cm2/s(2–10 keV)].

(b) The initial position error is 10 km, and initial velocity error is5 m/s respectively.

(c) The initial values of P0 (the initial error covariance), Q (pro-cess noise covariance matrix) and R (measurement noise co-variance matrix) are as following.

P0 = [10 0002 10 0002 10 0002 52 52 52]

Q = [2e–2 2e–2 2e–2 2e–4 2e–4 2e–4]R = [

1092 3255 3442]4.2. Results analysis

In order to illustrate the accuracy and superiority of the ODmethod based on X-ray pulsar for deep space exploration espe-cially when the orbit initial errors and orbit mutation exist, weinvestigate the proposed X-ray pulsar OD method from the follow-ing different aspects.

(1) The impact of initial errors on position and velocity estima-tion.

In order to elucidate the impact of initial error on the ODaccuracy based on orbit dynamic model, we calculate the stepestimated error between the true orbit and estimated orbit ob-tained from orbit dynamic model at different initial error. Theresults are shown in Table 3. When the initial velocity and ini-tial position errors are zero, the step estimated error is smaller.When the initial errors exist (the initial position error is equal to10 km and the initial velocity error is equal to 5 m/s, respectively),the step estimated error is greater than initial error. From the re-sults, we can draw clearly that the orbit dynamic model exertsless impact on the estimation of orbit, while the initial orbit er-rors are the main influencing factor, and pose a great impact onthe OD accuracy. In addition, because the filter algorithm is notused during the orbit estimation, the estimated errors of next timebecome bigger due to error accumulation even if there are no ini-tial errors. Therefore, the filter algorithm plays an important role

D. Feng et al. / Aerospace Science and Technology 32 (2014) 35–41 39

Table 3Step estimated errors under different initial errors.

Probe Orbit No error With error

Position Velocity Position Velocity

Phoenix Cruise phase 0.0162 m 5.3832 × 10−5 m/s 2.2517 × 104 m 8.6602 m/sDeep Impact Cruise phase 0.0393 m 1.3710 × 10−4 m/s 2.2517 × 104 m 8.6603 m/sVoyager 1 Encounter with Jupiter phase 0.6173 m 0.0032 m/s 2.2517 × 104 m 8.6620 m/s

Encounter with Saturn phase 0.9586 m 0.0041 m/s 2.2517 × 104 m 8.6614 m/s

Fig. 4. OD errors at cruise phase of Phoenix with different filtering algorithms.

Fig. 5. OD errors at cruise phase of Deep Impact with different filtering algorithms.

in reducing the estimated errors whether the initial errors exist ornot.

(2) The impact of filtering algorithm on the OD accuracy.This subsection reports effects of different filter algorithms

(EKF, UKF and AUKF) on the OD accuracy of the stable and smoothorbit phase when the initial errors exist. We select the cruise phaseof Phoenix and Deep Impact in OD method, and their orbit charac-teristics are relatively stable.

Fig. 4 shows the OD results using EKF, UKF and AUKF algorithmsin the cruise phase of Phoenix. Fig. 5 shows the OD results us-ing the same three algorithms in the cruise phase of Deep Impact.All performance curves were obtained with 10 km initial positionerrors and 5 m/s initial velocity errors. As the results shown inFigs. 4 and 5, the AUKF-based X-ray pulsar OD system providesthe highest OD accuracy. As the detailed curves demonstrated, ODestimated errors are convergent and relatively stable using UKFand AUKF algorithms, while OD estimated errors are divergent andfluctuating using EKF algorithm. One reason is that EKF algorithmis sensitive to initial errors. Another reason is that there is a bigtruncation error of EKF algorithm, which results from the fact thatEKF is suitable to linear system. In our paper the orbit dynamicmodel and measurement equation are nonlinear. A linear process

is needed when EKF filtering is used, in which case the truncationerror occurs.

(3) The impact of orbit characteristics on OD accuracy.Besides the initial orbit errors, orbit mutation is another es-

sential reason which affects the OD accuracy greatly. This partdemonstrates the impacts of different filter algorithms (EKF, UKFand AUKF) on the OD performances while the orbit mutationsexist. Figs. 6 and 7 show the OD results in the phases of Voy-ager 1 encountering with Jupiter and Saturn respectively. As theresults shown in Figs. 6 and 7, the AUKF-based X-ray pulsar ODsystem provides the highest OD accuracy especially when the or-bit changes greatly encountering other planets. The detailed curvesin Figs. 6 and 7 show that the OD errors based on EKF and UKFalgorithm are no convergence. It is clearly that the position and ve-locity estimation errors are small at initial time while the positionand velocity errors begin to diverge and the estimated positiongoes far beyond the real orbit when the orbits change significantly.The position estimation errors even reach to the order of 107.The results curves also suggest that OD errors, which are basedon AUKF, are convergent although the estimated errors are largerwhen Voyager 1 encounter with Jupiter and Saturn. The OD errorsare smaller than the errors based on UKF and EKF. More impor-

40 D. Feng et al. / Aerospace Science and Technology 32 (2014) 35–41

Fig. 6. OD errors at the phase of Voyager 1 encountering with Jupiter.

Fig. 7. OD errors at the phase of Voyager 1 encountering with Saturn.

tantly, the estimated errors based on AUKF are able to converge,especially when orbit mutations exist. The OD estimated errors aredivergent or fluctuating using EKF algorithm. The reasons are thatEKF is sensitive to orbit mutation and that there is a big truncationerror of EKF algorithm.

In summary, it is clear that the performance of the OD methodbased on AUKF and X-ray pulsar is superior. The OD based on AUKFcan reduce not only the impact of the initial errors on OD accuracy,but also the impact of the orbit mutation on OD accuracy.

5. Conclusions

This paper studies the impact of the initial errors and orbitcharacteristics on OD accuracy in deep space exploration using X-ray pulsar. To reduce the estimation errors of position and velocity,we propose the OD method based on X-ray pulsar and AUKF al-gorithm. Three different filtering methods have been studied forautonomous OD using X-ray pulsar and their performances havebeen compared for orbit estimation.

The proposed OD method is tested with data from horizonsystem, and the simulation results demonstrate that AUKF-basedX-ray pulsar OD method reduces the estimation errors, which re-sult from initial errors, orbit mutation, long filtering cycle and orbitdynamic model, with fast and stable convergence effectively. AUKFalgorithm yields the best OD accuracy, and estimation errors arestable and convergent especially when orbit characteristics changeseriously. Therefore, the autonomous OD system based on AUKFusing X-ray pulsar is feasible in deep space exploration. If it is reli-able in practical application, further studies should be carried out,such as the development of the onboard OD system for spacecraft.In addition, though a long observation time is required, the pro-posed X-ray pulsar-based OD system doesn’t rely on the ground

stations and has great anti-interference capability. Hence, thereis good complementarity between the proposed OD system andother approaches, such as optical navigation, star sensor naviga-tion and inertial navigation. Accordingly, the coordination of theproposed X-ray pulsar-based OD system together with other navi-gation methods can be adapted to the deep space autonomous ODsystem to provide enhanced OD accuracy.

Acknowledgements

All of the authors would like to thank you for the reviewers’useful comments and suggestions on our manuscript. This workwas supported by the National Natural Science Foundation of Chinaunder Grant Nos. 61203202, 61074194 and 61201298, and NPUFoundation for Fundamental Research (NPU-FFR-JC201206).

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